negation of a null sequence I have that a sequence $\{a_n\}$ is null $\Leftrightarrow \forall \epsilon >0, \exists X$ such that $$|a_n| < \epsilon  \ \forall n > X.$$
I want to give a definition when a sequence is not null, this is my approach, if $P_n(X) = |a_n| < \epsilon  \ \forall n > X$. Then, we have that:
$$\begin{align*}
\{a_n\} \ \mbox{is not null} \ \Leftrightarrow \neg[\forall \epsilon >0, \exists X P_n(X)] \\ 
 \Leftrightarrow \exists \epsilon >0 \ \neg \exists X P_n(X) \\ 
 \Leftrightarrow \exists \epsilon >0 \ \forall X \neg  P_n(X) 
\end{align*}$$
But I'm kinda confused on how to work out $\neg P_n(X)$. My attempt is the following: $$\begin{align*}
\neg P_n(X) \Rightarrow \neg[|a_n| < \epsilon  \ \forall n > X] \\ 
\neg P_n(X) \Rightarrow \exists n > X \ \mbox{with}\ |a_n| > \epsilon\\ 
\end{align*}$$
So, the whole statement would be
$$\exists \epsilon >0 \ \forall X [\exists n > X \ \mbox{with}\ |a_n| > \epsilon]$$
Is this correct and make sense, thanks.
 A: Yes, your work is all correct. Except a minor issue: the opposite of $|a_n| < \epsilon$ is $|a_n| \ge \epsilon$, not $|a_n| > \epsilon$.
Instead of writing $P_n(X)$ as $|a_n| < \epsilon \; \forall n > X$, you may have found it clearer to write it as $\forall n > X \; |a_n| < \epsilon$. Then your entire statement would have been
$$
(\forall\epsilon > 0) \; (\exists X \in \mathbb{N}) \; (\forall n > X) \; : \; |a_n| < \epsilon
$$
which is very straightforward to negate, as you have done.
Some mathematicians have a habit of putting the quantifier ($\forall n$) after the statement in question, for instance they might say: 
"there exists $M$, such that $|a_n| < M$ for all $n$." The problem with such statements is that the syntax doesn't indicate whether they mean $\exists M \; \forall n \; |a_n| < M$, or $\forall n \; \exists M \; |a_n| < M$, which are two very different statements. You have to figure out where the $\forall n$ belongs from the context. In general I think this is a bad habit that should be avoided.
